= Solution
Let $q=\mathbf k\cdot\widehat{\mathbf n}$ and retain the negative-spatial-perturbation convention of the preceding part. Since $h=h_S$, the isotropic pieces of the supplied <Fourier mode> contraction cancel:
$$
\left[\frac13h'\delta_{ij}
+\left(\widehat k_i\widehat k_j-\frac13\delta_{ij}\right)h'_S\right]
\widehat n_i\widehat n_j
=\frac{q^2}{k^2}h'.
$$
For $h=A(\mathbf k)\tau^2k^2$, the factor $1/2$ in the line-of-sight integral cancels the derivative's factor two. Thus
$$
\frac{\delta T}{T}
=\sum_{\mathbf k}A(\mathbf k)q^2
\int_{\tau_{\rm dec}}^{\tau_0}\tau e^{iq\tau}d\tau.
$$
An antiderivative convenient even at $q=0$ is
$$
\frac{d}{d\tau}\left[(1-iq\tau)e^{iq\tau}\right]
=q^2\tau e^{iq\tau}.
$$
It follows directly, or by two <integrations by parts>, that
$$
\boxed{\frac{\delta T}{T}
=-\left[\sum_{\mathbf k}i(\mathbf k\cdot\widehat{\mathbf n})
A(\mathbf k)\tau e^{i\mathbf k\cdot\widehat{\mathbf n}\tau}
\right]_{\tau_{\rm dec}}^{\tau_0}
+\left[\sum_{\mathbf k}A(\mathbf k)
e^{i\mathbf k\cdot\widehat{\mathbf n}\tau}
\right]_{\tau_{\rm dec}}^{\tau_0}.}
$$
The source sign and both endpoint coefficients are fixed by the metric convention; no division by $q$ is needed.
To interpret the large-angle <Cosmic microwave background anisotropy>, restore the observer-centered propagation phase $e^{-iq\tau_0}$, which can equivalently be absorbed into the definition of each <Fourier mode> amplitude. Put $D=\tau_0-\tau_{\rm dec}$. One mode contributes
$$
\Theta_{\mathbf k}(\widehat{\mathbf n})
=A(1-iq\tau_0)-A(1-iq\tau_{\rm dec})e^{-iqD}.
$$
The first term is a local angular monopole plus a dipole, so it is removed when discussing observed angular multipoles $\ell\ge2$. For modes outside the horizon at decoupling, $k\tau_{\rm dec}\ll1$, the higher-multipole emission signal is dominated by
$$
\boxed{\Theta_{\mathbf k,\ell\ge2}\simeq
-A(\mathbf k)e^{-i\mathbf k\cdot\widehat{\mathbf n}D}.}
$$
Its transfer amplitude has no leading positive power of $k$: the two explicit powers in $h$ have canceled in the integral. Projection onto the sky supplies <Spherical Bessel functions> $j_\ell(kD)$, with modes of order $kD\sim\ell$ contributing to each angular scale. This is the large-angle <Sachs-Wolfe effect>; the emission velocity-like correction is relatively of order $k\tau_{\rm dec}$. For wavelengths much larger than the entire observed region, $kD\ll1$, removing the monopole and dipole leaves a signal starting at order $A(kD)^2$, rather than a finite observable anisotropy from a spatially uniform mode.
The stochastic scale dependence of $A(\mathbf k)$ is not specified by the deterministic metric solution. If one additionally assumes a scale-invariant primordial spectrum, $k^3P_A(k)=\text{constant}$, then
$$
C_\ell\ \propto\ \int_0^\infty\frac{dk}{k}\,j_\ell(kD)^2
=\frac1{2\ell(\ell+1)}
$$
for $\ell\ge1$, giving the familiar large-angle <Sachs-Wolfe plateau>, $\ell(\ell+1)C_\ell\simeq\text{constant}$, for the measured $\ell\ge2$ multipoles. A tilted primordial spectrum changes that trend. The full numerical temperature transfer also includes intrinsic last-scattering and Doppler terms, omitted from the stipulated metric-only integral.
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